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Simplifying Radical Expressions

Simplify.

20x4y3

−454x6y3

x2−14x+49

(x−8)4

Simplify. (Assume all variable expressions are nonzero.)

100x2y4

36a6b2

8a2b4

72x4yz6

10x150x7y4

−5n225m10n6

48x6y3z23

270a10b8c33

a3b564c63

a2632b5c105

The period T in seconds of a pendulum is given by the formula T=2πL32 where L represents the length in feet of the pendulum. Calculate the period of a pendulum that is 212 feet long. Give the exact answer and the approximate answer to the nearest hundredth of a second.

The time in seconds an object is in free fall is given by the formula t=s4 where s represents the distance in feet the object has fallen. How long does it take an object to fall 28 feet? Give the exact answer and the approximate answer to the nearest tenth of a second.

Find the distance between (−5, 6) and (−3,−4).

Find the distance between (23,−12) and (1,−34).

Determine whether or not the three points form a right triangle. Use the Pythagorean theorem to justify your answer.

(−4,5), (−3,−1), and (3,0)

(−1,−1), (1,3), and (−6,1)

Adding and Subtracting Radical Expressions

Simplify. Assume all radicands containing variables are nonnegative.

72+52

815−215

143+52−53−62

22ab−5ab+7ab−2ab

7x−(3x+2y)

(8yx−7xy)−(5xy−12yx)

(35+26)+(85−36)

(433−123)−(533−2123)

(2−10x+3y)−(1+210x−6y)

(3aab23+6a2b3)+(9aab23−12a2b3)

45+12−20−75

24−32+54−232

23x2+45x−x27+20x

56a2b+8a2b2−224a2b−a18b2

5y4x2y−(x16y3−29x2y3)

(2b9a2c−3a16b2c)−(64a2b2c−9ba2c)

216x3−125xy3−8x3

128x33−2x543+32x33

8x3y3−2x8y3+27x3y3+xy3

27a3b3−38ab33+a64b3−ba3

Calculate the perimeter of the triangle formed by the following set of vertices: {(−3,−2),(−1,1),(1,−2)}.

Calculate the perimeter of the triangle formed by the following set of vertices: {(0,−4),(2,0),(−3,0)}.

Multiplying and Dividing Radical Expressions

Multiply.

6⋅15

(42)2

2(2−10)

(5−6)2

(5−3)(5+3)

(26+3)(2−53)

(a−5b)2

3xy(x−2y)

3a23⋅18a3

49a2b3⋅7a2b23

Divide. Assume all variables represent nonzero numbers and rationalize the denominator where appropriate.

729

104864

55

152

326

2+510

183x

23x6xy

13x23

5ab25a2b3

5xz249x2y2z3

18x4y2z5

9x2y81xy2z35

27ab315a4bc25

15−3

32+1

−362−10

xyx−y

2−62+6

a+ba−b

The base of a triangle measures 26 units and the height measures 315 units. Find the area of the triangle.

If each side of a square measures 5+210 units, find the area of the square.

Rational Exponents

Express in radical form.

111/2

22/3

x3/5

a−4/5

Write as a radical and then simplify.

161/2

721/2

82/3

321/3

(19)3/2

(1216)−1/3

Perform the operations and simplify. Leave answers in exponential form.

61/2⋅63/2

31/3⋅31/2

65/263/2

43/441/4

(64x6y2)1/2

(27x12y6)1/3

(a4/3a1/2)2/5

(16x4/3y2)1/2

56x3/4y3/214x1/2y2/3

(4a4b2/3c4/3)1/22a2b1/6c2/3

(9x−4/3y1/3)−3/2

(16x−4/5y1/2z−2/3)−3/4

Perform the operations with mixed indices.

y⋅y25

y3⋅y35

y23y5

y23

Solving Radical Equations

Solve.

2x+3=13

3x−2=4

x−5+4=8

5x+3+7=2

4x−3=2x+15

8x−15=x

x−1=13−x

4x−3=2x−3

x+5=5−x

x+3=3x−1

2(x+1)−x+2=1

6−x+x−2=2

3x−2+x−1=1

9−x=x+16−1

4x−33=2

x−83=−1

x(3x+10)3=2

2x2−x3+4=5

3(x+4)(x+1)3=5x+373

3x2−9x+243=(x+2)23

y1/2−3=0

y1/3+3=0

(x−5)1/2−2=0

(2x−1)1/3−5=0

(x−1)1/2=x1/2−1

(x−2)1/2−(x−6)1/2=2

(x+4)1/2−(3x)1/2=−2

(5x+6)1/2=3−(x+3)1/2

Solve for g: t=2sg.

Solve for x: y=x+43−2.

The period in seconds of a pendulum is given by the formula T=2πL32 where L represents the length in feet of the pendulum. Find the length of a pendulum that has a period of 112 seconds. Find the exact answer and the approximate answer rounded off to the nearest tenth of a foot.

The outer radius of a spherical shell is given by the formula r=3V4π3+2 where V represents the inner volume in cubic centimeters. If the outer radius measures 8 centimeters, find the inner volume of the sphere.

The speed of a vehicle before the brakes are applied can be estimated by the length of the skid marks left on the road. On dry pavement, the speed v in miles per hour can be estimated by the formula v=26d, where d represents the length of the skid marks in feet. Estimate the length of a skid mark if the vehicle is traveling 30 miles per hour before the brakes are applied.

Find the real root of the function defined by f(x)=x−33+2.

Complex Numbers and Their Operations

Write the complex number in standard forma+bi.

5−−16

−−25−6

3+−810

−12−46

Perform the operations.

(6−12i)+(4+7i)

(−3+2i)−(6−4i)

(12−i)−(34−32i)

(58−15i)+(32−23i)

(5−2i)−(6−7i)+(4−4i)

(10−3i)+(20+5i)−(30−15i)

4i(2−3i)

(2+3i)(5−2i)

(4+i)2

(8−3i)2

(3+2i)(3−2i)

(−1+5i)(−1−5i)

2+9i2i

i1−2i

4+5i2−i

3−2i3+2i

10−5(2−3i)2

(2−3i)2−(2−3i)+4

(11−i)2

(1+2i3i)2

−8(3−−4)

(1−−18)(3−−2)

(−5−−10)2

(1−−2)2−(1+−2)2

Show that both −5i and 5i satisfy x2+25=0.

Show that both 1−2i and 1+2i satisfy x2−2x+5=0.

Answers

−11

|xy|

5

xy

f(−1)=3; f(6)=4

[−25,∞)

523

−943

−9

−265

2x2|y|5y

|x−7|

10xy2

2a2b2

50x4y26x

2x2y6z23

abb234c2

π54 seconds; 1.76 seconds

226 units

Right triangle

122

93−2

4x−2y

115−6

1−310x+9y

5−33

−x3+55x

12xyy

4x3−5xy3

2xy3

4+213 units

310

2−25

22

a−10ab+25b

3a23

22

5

64

63xx

9x33x

35x2yz37xy

3xy3x4y3z25z

5+32

6+15

−2+3

910 square units

11

x35

4

4

1/27

36

6

8x3y

a1/3

4x1/4y5/6

x227y1/2

y910

y715

25

21

9

4

4

7

1

114

−4,23

−5,53

9

9

1

12

g=2st2

18π2 feet; 1.8 feet

37.5 feet

5−4i

310+25i

10−5i

−14+12i

3+i

12+8i

15+8i

13

92−i

35+145i

35+60i

12i

42+2i6

−15+102

Answer may vary

Sample Exam

Simplify. (Assume all variables are positive.)

5x121x2y4

2xy2−64x6y93

Calculate the distance between (−5,−3) and (−2,6).

The time in seconds an object is in free fall is given by the formula t=s4 where s represents the distance in feet that the object has fallen. If a stone is dropped into a 36-foot pit, how long will it take to hit the bottom of the pit?

Perform the operations and simplify. (Assume all variables are positive and rationalize the denominator where appropriate.)

150xy2−218x3+y24x+x128x

316x3y23−(2x250y23−54x3y23)

22(2−36)

(10−5)2

62+3

2x2xy

18xy2z45

Simplify: 813/4.

Express in radical form: x−3/5.

Simplify. Assume all variables are nonzero and leave answers in exponential form.

(81x4y2)−1/2

(25a4/3b8)3/2a1/2b

Solve.

x−5=1

5x−23+6=4

52x+5−2x=11

4−3x+2=x

2x+5−x+3=2

The time in seconds an object is in free fall is given by the formula t=s4 where s represents the distance in feet that the object has fallen. If a stone is dropped into a pit and it takes 4 seconds to reach the bottom, how deep is the pit?

The width in inches of a container is given by the formula w=4V32+1 where V represents the inside volume in cubic inches of the container. What is the inside volume of the container if the width is 6 inches?